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270       XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

each  s<=G,   there  exists  an  integer  n   such   that   l^n^m  and

(If 77 : G - G/H is the canonical mapping, observe that /x( 7r(U)) > 0 and that the

measures of the sets sj  7r(U) are all equal.)

5.   Let G be a locally compact group and ^ a right Haar measure on G.

(a)    In order that a positive measure a ^ 0 on G, with support S, should be such that
S is a closed subgroup of G and the measure induced on S by a a right Haar measure
on S, it is necessary and sufficient that 5Cs)a = a for all s e S. The set of elements
/ e G such that 5(r)a = a is then equal to S.

(b)    Let F denote the set of positive measures a on G satisfying the condition of (a)
above. Show that T is vaguely closed in M+(G)- {0}. If /e JT+(G) is such that

f(e) > 0, show that the set F/ of measures a e T such that m doc(s)  I is vaguely
compact (prove that sup a(K) < + oo for every compact subset K of G). Deduce that

a e r/

the mapping ai>(a(/), a/a(/)) is a homeomorphism of T onto the product space
R*+ xiy

(c)    For each a e F, let H = Supp(a) : it is a closed subgroup of G. Let d be a right-
invariant distance on G such that d(x, y)^l for all x, y in G (12.9.1). Let (Kn) be an
increasing sequence of compact sets, covering G and such that Kn is contained in the
interior of Kra+1 for each n (3.18.3). If h is the HausdorfT distance on G corresponding
to d (Section 3.16, Problem 3), then for any two nonempty closed sets M, N in G we
define   hn(M, N) = 1   if   either   M r\ Kn   or   N n Kn   is   empty, and /rn(M, N) 
h(M r\ Kn, N n Kn) otherwise. Endow the set fHG) of nonempty closed subsets of
G with the topology defined by the pseudo-distances hn. Show that the mapping
a h-> H is a homeomorphism of F/ (endowed with the vague topology) onto the set
2 of closed subgroups of G, endowed with the topology induced by that of

6. With the notation of Problem 5, consider the subset F of F consisting of measures
a e F such that Ha is unimodular. This is also the set of measures a e F such that
a(/) = a(/) for all /e Jf(G), and is therefore vaguely closed in F. For each a e F,
put Qa = G/H; then there exists a relatively G-invariant measure ^ on Q such that

f f(x)d^(x)= f   d^(x) (  f(xs)doc(s)

JG                            JQa                 ^Ho

for every /e ^T(G) (where x is the coset xUa) (Problem 2).

(a)    If a 6 F and /e JT(G), put /(*) = f   f(xs) da(s). Show that the mapping

JH<X

ai > ll/all is vaguely continuous, and deduce that the mapping ai > ||jiia|| is lower semi-
continuous with respect to the vague topology.

(b)    Let g ^ 0 be a real-valued jit-integrable function, and let F(#) be the set of

measures a e F such that I g(xs) du(s) g: 1 for all x e G. Show that the mapping
ai>> ll/i-all of F(^) into R is vaguely continuous, (It is enough to show that the map-
ping is upper semicontinuous. Let h e JT+(G) be such that I \g(x)  h(x)\ dp.(x) ^ e,

and let K = Supp(/z). If TT^ : G -> Q^ is the canonical mapping, show that
fiff(Q0 - 77>(K)) S  for all f$ E T(g). On the other hand, consider a f unction /eJf+(G)

such that f f(xs) da(s) <i 1 for all s e G and f f(xs) da(s) = 1 for all s e K (use Prob-

JG                                                              JG

lem 2(a)). If Ue is the set of measures j8 e F(^) such that f f(xsdft(s)> 1 - e for
all x eK, show that (1 - e)^(ir/K)) < ||f*.||.) 6).pology ^~4, v (use (e)). Give an example of a